scholarly journals On the steady-state nearly resonant waves

2016 ◽  
Vol 794 ◽  
pp. 175-199 ◽  
Author(s):  
Shijun Liao ◽  
Dali Xu ◽  
Michael Stiassnie
Keyword(s):  
Steady State ◽  
Deep Water ◽  
Water Waves ◽  
Wave Equations ◽  
Linear Part ◽  
Wave Frequency ◽  
Analytic Approximation ◽  
Approximation Approach ◽  
Full Wave ◽  
Resonant Waves

The steady-state nearly resonant water waves with time-independent spectrum in deep water are obtained from the full wave equations for inviscid, incompressible gravity waves in the absence of surface tension by means of a analytic approximation approach based on the homotopy analysis method (HAM). Our strategy is to mathematically transfer the steady-state nearly resonant wave problem into the steady-state exactly resonant ones. By means of choosing a generalized auxiliary linear operator that is a little different from the linear part of the original wave equations, the small divisor, which is unavoidable for nearly resonant waves in the frame of perturbation methods, is avoided, or moved far away from low wave frequency to rather high wave frequency with physically negligible wave energy. It is found that the steady-state nearly resonant waves have nothing fundamentally different from the steady-state exactly resonant ones, from physical and numerical viewpoints. In addition, the validity of this HAM-based analytic approximation approach for the full wave equations in deep water is numerically verified by means of the Zakharov’s equation. A thought experiment is discussed, which suggests that the essence of the so-called ‘wave resonance’ should be reconsidered carefully from both of physical and mathematical viewpoints.

Steady-state resonance of multiple wave interactions in deep water

2014 ◽  
Vol 742 ◽  
pp. 664-700 ◽  
Author(s):  
Zeng Liu ◽  
Shi-Jun Liao
Keyword(s):  
Steady State ◽  
Parameter Space ◽  
Gravity Waves ◽  
Deep Water ◽  
Wave Energy ◽  
Two Dimensions ◽  
Physical Parameters ◽  
Wave System ◽  
State Resonance ◽  
Resonant Waves

AbstractThe steady-state resonance of multiple surface gravity waves in deep water was investigated in detail to extend the existing results due to Liao (Commun. Nonlinear Sci. Numer. Simul., vol. 16, 2011, pp. 1274–1303) and Xu et al. (J. Fluid Mech., vol. 710, 2012, pp. 379–418) on steady-state resonance from a quartet to more general and coupled resonant quartets, together with higher-order resonant interactions. The exact nonlinear wave equations are solved without assumptions on the existence of small physical parameters. Multiple steady-state resonant waves are obtained for all the considered cases, and it is found that the number of multiple solutions tends to increase when more wave components are involved in the resonance sets. The topology of wave energy distribution in the parameter space is analysed, and it is found that the steady-state resonant waves indeed form a continuum in the parameter space. The significant roles of the near-resonance and nonlinearity were also revealed. It is found that all of the near-resonant components as a whole contain more and more wave energy, as the wave patterns tend from two dimensions to one dimension, or as the nonlinearity of the steady-state resonant wave system increases. In addition, the linear stability of the steady-state resonant waves is analysed. It is found that the steady-state resonant waves are stable, as long as the disturbance does not resonate with any components of the basic wave. All of these findings are helpful to enrich and deepen our understanding about resonant gravity waves.

scholarly journals Experimental study of particle trajectories below deep-water surface gravity wave groups

2019 ◽  
Vol 879 ◽  
pp. 168-186 ◽  
Author(s):  
T. S. van den Bremer ◽  
C. Whittaker ◽  
R. Calvert ◽  
A. Raby ◽  
P. H. Taylor
Keyword(s):  
Gravity Wave ◽  
Deep Water ◽  
Water Waves ◽  
Wave Equations ◽  
Stokes Drift ◽  
Surface Gravity ◽  
Return Flow ◽  
Surface Gravity Wave ◽  
Wave Groups ◽  
Wave Induced

Owing to the interplay between the forward Stokes drift and the backward wave-induced Eulerian return flow, Lagrangian particles underneath surface gravity wave groups can follow different trajectories depending on their initial depth below the surface. The motion of particles near the free surface is dominated by the waves and their Stokes drift, whereas particles at large depths follow horseshoe-shaped trajectories dominated by the Eulerian return flow. For unidirectional wave groups, a small net displacement in the direction of travel of the group results near the surface, and is accompanied by a net particle displacement in the opposite direction at depth. For deep-water waves, we study these trajectories experimentally by means of particle tracking velocimetry in a two-dimensional flume. In doing so, we provide visual illustration of Lagrangian trajectories under groups, including the contributions of both the Stokes drift and the Eulerian return flow to both the horizontal and the vertical Lagrangian displacements. We compare our experimental results to leading-order solutions of the irrotational water wave equations, finding good agreement.

The effect of the induced mean flow on solitary waves in deep water

1998 ◽  
Vol 355 ◽  
pp. 317-328 ◽  
Author(s):  
T. R. AKYLAS ◽  
F. DIAS ◽  
R. H. J. GRIMSHAW
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Deep Water ◽  
Water Waves ◽  
Wave Equations ◽  
Soliton Solutions ◽  
Phase Speed ◽  
Mean Flow ◽  
Nls Equation ◽  
Envelope Soliton

Two branches of gravity–capillary solitary water waves are known to bifurcate from a train of infinitesimal periodic waves at the minimum value of the phase speed. In general, these solitary waves feature oscillatory tails with exponentially decaying amplitude and, in the small-amplitude limit, they may be interpreted as envelope-soliton solutions of the nonlinear Schrödinger (NLS) equation such that the envelope travels at the same speed as the carrier oscillations. On water of infinite depth, however, based on the fourth-order envelope equation derived by Hogan (1985), it is shown that the profile of these gravity–capillary solitary waves actually decays algebraically (like 1/x2) at infinity owing to the induced mean flow that is not accounted for in the NLS equation. The algebraic decay of the solitary-wave tails in deep water is confirmed by numerical computations based on the full water-wave equations. Moreover, the same behaviour is found at the tails of solitary-wave solutions of the model equation proposed by Benjamin (1992) for interfacial waves in a two-fluid system.

Finite amplitude steady-state wave groups with multiple near resonances in deep water

2017 ◽  
Vol 835 ◽  
pp. 624-653 ◽  
Author(s):  
Z. Liu ◽  
D. L. Xu ◽  
S. J. Liao
Keyword(s):  
Steady State ◽  
Linear Operator ◽  
Deep Water ◽  
Wave Equations ◽  
Finite Amplitude ◽  
Wave Groups ◽  
Homogeneous Solutions ◽  
State Wave ◽  
Resonant Interactions ◽  
Auxiliary Linear Operator

In this paper, finite amplitude steady-state wave groups with multiple nearly resonant interactions in deep water are investigated theoretically. The nonlinear water wave equations are solved by the homotopy analysis method (HAM), which imposes no constraint on either the number or the amplitude of the wave components, to resolve the small-divisor problems caused by near resonances. A new kind of auxiliary linear operator in the framework of the HAM is proposed to transform the small divisors associated with the non-trivial nearly resonant components to singularities associated with the exactly resonant ones. Primary components, exactly resonant components together with nearly resonant components are considered as the initial non-trivial components, since all of them are homogeneous solutions to the auxiliary linear operator. For wave groups with weak nonlinearity, the energy transfer between nearby nearly resonant components is remarkable. As the nonlinearity increases, the number of steady-state wave groups increases as more components join the near resonance. This indicates that the probability of existence of steady-state resonant waves increases with the nonlinearity of wave groups. The frequency band broadens and spectral asymmetry becomes more and more pronounced. The amplitude of each component may either increase or decrease with the nonlinearity of wave groups, while the amplitude of the whole wave group increases continuously and finite amplitude wave groups are obtained. This work shows the wide existence of steady-state waves when multiple nearly resonant interactions are considered.

On the Exact Soliton Solutions of Some Nonlinear PDE by Tan-Cot Method

2018 ◽  
Vol 5 (1) ◽  
pp. 31-36
Author(s):  
Md Monirul Islam ◽  
Muztuba Ahbab ◽  
Md Robiul Islam ◽  
Md Humayun Kabir
Keyword(s):  
Solitary Wave ◽  
Acoustic Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Rectangular Channel ◽  
Soliton Solutions ◽  
Boussinesq Equations ◽  
Travelling Wave ◽  
Ion Acoustic Waves ◽  
Analytical System

For many solitary wave applications, various approximate models have been proposed. Certainly, the most famous solitary wave equations are the K-dV, BBM and Boussinesq equations. The K-dV equation was originally derived to describe shallow water waves in a rectangular channel. Surprisingly, the equation also models ion-acoustic waves and magneto-hydrodynamic waves in plasmas, waves in elastic rods, equatorial planetary waves, acoustic waves on a crystal lattice, and more. If we describe all of the above situation, we must be needed a solution function of their governing equations. The Tan-cot method is applied to obtain exact travelling wave solutions to the generalized Korteweg-de Vries (gK-dV) equation and generalized Benjamin-Bona- Mahony (BBM) equation which are important equations to evaluate wide variety of physical applications. In this paper we described the soliton behavior of gK-dV and BBM equations by analytical system especially using Tan-cot method and shown in graphically. GUB JOURNAL OF SCIENCE AND ENGINEERING, Vol 5(1), Dec 2018 P 31-36

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

2020 ◽  
Vol 11 (1) ◽  
pp. 93-100
Author(s):  
Vina Apriliani ◽  
Ikhsan Maulidi ◽  
Budi Azhari
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Internal Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Innovative Methods ◽  
De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations. 

scholarly journals Bound Coherent Structures Propagating on the Free Surface of Deep Water

Fluids ◽  
2021 ◽  
Vol 6 (3) ◽  
pp. 115
Author(s):  
Dmitry Kachulin ◽  
Sergey Dremov ◽  
Alexander Dyachenko
Keyword(s):  
Free Surface ◽  
Deep Water ◽  
Coherent Structures ◽  
Water Waves ◽  
Nonlinear Models ◽  
Ideal Incompressible Fluid ◽  
Equation Model ◽  
System Of Nonlinear Equations ◽  
Potential Flows ◽  
Deep Water Waves

This article presents a study of bound periodically oscillating coherent structures arising on the free surface of deep water. Such structures resemble the well known bi-soliton solution of the nonlinear Schrödinger equation. The research was carried out in the super-compact Dyachenko-Zakharov equation model for unidirectional deep water waves and the full system of nonlinear equations for potential flows of an ideal incompressible fluid written in conformal variables. The special numerical algorithm that includes a damping procedure of radiation and velocity adjusting was used for obtaining such bound structures. The results showed that in both nonlinear models for deep water waves after the damping is turned off, a periodically oscillating bound structure remains on the fluid surface and propagates stably over hundreds of thousands of characteristic wave periods without losing energy.

Steep gravity waves in water of arbitrary uniform depth

1977 ◽  
Vol 286 (1335) ◽  
pp. 183-230 ◽  
Keyword(s):  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Perturbation Parameter ◽  
Intermediate Energy ◽  
Finite Amplitude ◽  
Wave Profile ◽  
Accurate Calculation ◽  
Theoretical Developments ◽  
Deep Water Waves

Modern applications of water-wave studies, as well as some recent theoretical developments, have shown the need for a systematic and accurate calculation of the characteristics of steady, progressive gravity waves of finite amplitude in water of arbitrary uniform depth. In this paper the speed, momentum, energy and other integral properties are calculated accurately by means of series expansions in terms of a perturbation parameter whose range is known precisely and encompasses waves from the lowest to the highest possible. The series are extended to high order and summed with Padé approximants. For any given wavelength and depth it is found that the highest wave is not the fastest. Moreover the energy, momentum and their fluxes are found to be greatest for waves lower than the highest. This confirms and extends the results found previously for solitary and deep-water waves. By calculating the profile of deep-water waves we show that the profile of the almost-steepest wave, which has a sharp curvature at the crest, intersects that of a slightly less-steep wave near the crest and hence is lower over most of the wavelength. An integration along the wave profile cross-checks the Padé-approximant results and confirms the intermediate energy maximum. Values of the speed, energy and other integral properties are tabulated in the appendix for the complete range of wave steepnesses and for various ratios of depth to wavelength, from deep to very shallow water.

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